Objects next to each other create a small white line.

I've been working on making some low-poly art and created triangles with the pen tool. These triangles were made by using the anchor points of other triangles. The results are great but there's just one thing that really bugs me. Between the objects, there is a very small white line which make the image look like it's not connected correctly. I assume this has something to do with the objects being vector objects. When I export to Photoshop its still visible.

Is there a possibility to fill those lines without screwing up the anchor points?

Try adding a small stroke the same color as the triangle to triangles showing those lines. This happens to me all the time. No bigger than half a point should work - may have to make adjustments for your art. You also add fill with the brush on a layer tucked behind.

Occasionally I have encountered this issue and solved it by methods Jonathon Ziegler suggested in another post in this thread.

But almost always I found the errors to be my mistake or snap not working entirely correctly.

Let's start with human error. You don't say, but I will guess your triangles are two different fill colors (if they were the same, you would have used Pathfinder > Add/Unite, right?). The first two things I look at are did I accidentally leave a stroke on either path? If I did I remove it.

Next is snap: I have guides, smart guides, and snap all turned on (usually). Sometimes snap just gets you "close", not exactly on the mark. Zoom in as much as your version of AI lets you and see if the anchors actually are right on top of each other. If not, move them.

Frankly, the second is more likely than the first these days... I've messed up so many times I check my settings before I draw. ;)

Now, if neither of these are the case, then I do as Jonathan suggests: I compensate for Adobe's limitations. Which brings to mind one last possibility: If one or both of your objects intentionally have a stroke it is a bit tougher, but solved similar to the snap problem by zooming in and adjusting your anchor points until the results look right.